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Journal of Function Spaces and Applications
Volume 2013 (2013), Article ID 982753, 16 pages
http://dx.doi.org/10.1155/2013/982753
Research Article

Estimates for Unimodular Multipliers on Modulation Hardy Spaces

1Department of Mathematics, Zhejiang Normal University, Jinhua 321004, China
2Department of Mathematics, University of Wisconsin-Milwaukee, Milwaukee, WI 53201, USA
3School of Science, Hangzhou Dianzi University, Hangzhou 310016, China

Received 23 November 2012; Accepted 23 January 2013

Academic Editor: Baoxiang Wang

Copyright © 2013 Jiecheng Chen et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

It is known that the unimodular Fourier multipliers are bounded on all modulation spaces for . We extend such boundedness to the case of all and obtain its asymptotic estimate as t goes to infinity. As applications, we give the grow-up rate of the solution for the Cauchy problems for the free Schrödinger equation with the initial data in a modulation space, as well as some mixed norm estimates. We also study the boundedness for the operator , for the case and Finally, we investigate the boundedness of the operator for and obtain the local well-posedness for the Cauchy problem of some nonlinear partial differential equations with fundamental semigroup .